0.1 A telecommunication system  (Page 13/15)

Our presentation focuses on solving problems using simple recursive (gradient)methods. Once the synchronization problems are correctly stated, techniques for their solution becomeobvious. With the exception of frame synchronization (which is approached via correlational methods)the problem of designing synchronizers is unified via one simple concept, that of the minimization (or maximization)of an appropriate performance function. [link] , [link] , and [link] contain details.

Equalization

When all is well in the digital receiver, there is no interaction between adjacent data values and allfrequencies are treated equally. In most real wireless systems (and many wired systems as well),however, the transmission channel causes multiple copies of the transmitted symbols, each scaled differently,to arrive at the receiver at different times. This intersymbol interference can garble the data. The channel may also attenuate different frequencies bydifferent amounts. Thus frequency selectivity can render the data indecipherable.

The solution to both of these problems is to build a filter in the receiver thatattempts to undo the effects of the channel. This filter, called an equalizer , cannot be fixed in advance by the system designer,however, because it must be different to compensate for different channel paths that are encountered when thesystem is operating. The problem of equalizer design can be stated as a simple optimization problem, that offinding a set of filter parameters to minimize an appropriate function of the error, given only the received data(and perhaps a training sequence). This problem is investigated in detail in Chapter [link] , where the same kinds of adaptive techniques used to solve the synchronization problemscan also be applied to solve the equalization problem.

Decisions and error measures

In analog systems, the transmitted waveform can attain any value, but in a digital implementationthe transmitted message must be one of a small number of values defined by the symbol alphabet.Consequently, the received waveform in an analog system can attain any value, but in a digital implementationthe recovered message is meant to be one of a small number of values from the source alphabet.Thus, when a signal is demodulated to a symbol and it is not amember of the alphabet, the difference between the demodulated value (called a soft decision) and the nearest element of the alphabet (the hard decision) can provide valuable information about the performance of thesystem.

To be concrete, label the signals at various points as shown in [link] :

• The binary input message $b(·)$ .
• The coded signal $w(·)$ is a discrete-time sequence drawn from a finite alphabet.
• The signal $m(·)$ at the output of the filter and equalizer is continuous valuedat discrete times.
• $Q\left\{m\right(·\left)\right\}$ is a version of $m(·)$ that is quantized to the nearest member of the alphabet.
• The decoded signal $\widehat{b}(·)$ is the final (binary) output of the receiver.

If all goes well and the message is transmitted, received, and decoded successfully,then the output should be the same as the input, although there may be some delay $\delta$ between the time of transmission and the time when theoutput is available. When the output differs from the message, then errors have occurred during transmission.